If Paul Churchland delves in vectorialism and vector coding, Patricia Churchland and Terrence Sejnowski take a sojourn into state-space physics and phase mathematics. This is to define a particular state a network is in by mapping it to a point. This has the disconcerting effect of lending specificity to the point thus mapped, and obviously misses out on the all important relationship between parameters, and sub-spaces. This leap into the state-space physics and phase space mathematics is further disconcerting, for we lag behind in our intuitive qualities in visualizing higher-dimensions spaces. Even if, we humans lag behind in visualizing the number of higher dimensions, we are capable of working this out through algebraic representations. Through this algebraic representation, the utility of the state-space description is found to be to our taste, with a caveat being the diagrams’ proclivity to slide into a basin of attraction or converge onto a point with increasing complexity. Irrespective of the higher-dimensional predicaments, Churchlands take the seriousness of connectionism right to the heart of their models, and despite their clinging on to representation, make it very clear that representation as treated in connectionism is a far cry from the way it is traditionally understood.
One thought on “Churchlands, Representational Disconcert via State-Space Physics & Phase-Space Mathematics”
Let a dynamic system be specified by a function f(t, *). If ‘a’ is a point in the phase space (a space in which all possible states of a system is represented, with each possible state of the system corresponding to one unique point in the phase space), so that a state of the system at a certain time, then f(0, a) = a and for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. An attractor, is then a subset A of the phase space characterized by three conditions:
a) A is forward invariant under f: if a is an element of A then so is f(t, a), ∀ t > 0.
b) There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that “enter A in the limit t → ∞”. More formally, B(A) is the set of all points b in the phase space with the following property:
For any open neighborhood N of A, there is a large positive constant T such that f(t,b) ∈ N ∀ ℜ t > T.
c) There is no proper subset of A having the first two properties.